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Results tagged with descriptive-set-theory
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user 15570
Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
4
votes
1
answer
236
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Is it possible for a separable metric and a non-separable metric to have the same Borel $\si...
I hope this is not too basic or obvious a question.
Let $d_1$ and $d_2$ be metrics on the same set $X$, with $d_1$ being separable and $d_2$ not being separable. Is it possible that $d_1$ and $d_2$ g …
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4
votes
1
answer
131
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Is every convex subset of a Borel-linearly ordered space measurable?
Let $(X,\Sigma)$ be a standard measurable space, and let $\,\preceq\,$ be a total order on $X$ with the property that $\,\{(x,y) \in X \times X: x \preceq y\} \in \Sigma \otimes \Sigma$.
Let $A \subs …
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9
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0
answers
255
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Is the inverse of a measurably parametrised family of bijections between standard Borel spac...
It is known that a measurable bijection $f \colon [0,1] \to [0,1]$ has a measurable inverse. (Here, all measurability is simply with respect to the Borel $\sigma$-algebra of $[0,1]$.)
Now fix an arbi …
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4
votes
0
answers
218
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Does it make sense to regard the graph of any function as being a "sort-of-null set"?
Following the nice answer to Do the Lebesgue-null sets cover "all the sets can naturally be regarded as sort-of-null sets"?, the particular situation that I am especially interested in (which is a kin …
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9
votes
0
answers
189
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For measure-preserving systems, is countable generatability of the invariant $\sigma$-algebr...
Let $X$ be a second countable Hausdorff topological space, let $T \colon X \to X$ be a Borel-measurable map, define the $\sigma$-algebra $\mathcal{I}=\{A \in \mathcal{B}(X) : T^{-1}(A)=A\}$, and for e …
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15
votes
1
answer
2k
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Is "conditioning to a sub-$\sigma$-algebra" a measurable operation?
Let $\mathcal{B}$ be the Borel $\sigma$-algebra of $[0,1]$, and let $\mathcal{M}$ be the set of probability measures on $([0,1],\mathcal{B})$, equipped with the evaluation $\sigma$-algebra $\ \sigma(\ …
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3
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0
answers
202
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Is the homeomorphism group of a Polish space a measurable group?
Let $X$ be a Polish space. Let $H(X)$ be the set of homeomorphisms $h \colon X \to X$, equipped with the "evaluation $\sigma$-algebra", namely $\sigma(h \mapsto h(x) : x \in X)$.
(Note that for any m …
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9
votes
1
answer
859
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Do the Lebesgue-null sets cover "all the sets can naturally be regarded as sort-of-null sets"?
Let $F$ be the set of bijective Borel-measurable functions $f \colon [0,1] \to [0,1]$ that preserve the Lebesgue measure.
Is it the case that for every non-Lebesgue-measurable set $A \subset …
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2
votes
0
answers
60
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Is there a nice characterisation of when a sub-$\sigma$-algebra induces a measurable conditi...
Preliminary notations: For a compact metrisable space $X$,
$\mathcal{B}(X)$ is the Borel $\sigma$-algebra on $X$.
$\overline{\mathcal{B}}(X)$ is the universal completion of $\mathcal{B}(X)$.
$\mathca …
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3
votes
1
answer
89
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Can the set of compact metrisable topologies naturally be equipped with the structure of a s...
Let $X$ be a compact metric space, and let $K_X$ be the set of non-empty closed subsets of $X$, equipped with the $\sigma$-algebra
$$ \mathcal{B}(K_X) \ := \ \sigma(\{C \in K_X : C \cap U = \emptyset\ …
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2
votes
0
answers
244
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Reference for Borel $\sigma$-algebra of topology of convergence in probability
I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before.
So I'm wondering if there are any papers/text …
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3
votes
1
answer
446
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"Strongly mutually singular" families of measures, and the set of ergodic measures
Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish].
Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume th …
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5
votes
1
answer
243
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Is the topology of weak+Hausdorff convergence Polish?
Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff convergence" …
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2
votes
1
answer
123
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Can convergence in distribution necessarily be realised by almost-sure convergence?
Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each …
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